The Wright function – hypergeometric representation and symbolical evaluation

The Wright function, which arises in the theory of the space-time fractional diffusion equation, is an interesting mathematical object which has diverse connections with other special and elementary functions. The Wright function provides a unified treatment of several classes of special functions, such as the Gaussian, Airy, Bessel, and Error functions, etc. The manuscript demonstrates an algorithm for symbolical representation in terms of finite sums of hypergeometric (HG) functions and polynomials. The HG functions are then represented by known elementary or other special functions, wherever possible. The algorithm is programmed in the open-source computer algebra system Maxima and can be used to for testing numerical algorithms for the evaluation of the Wright function.